# Difference between revisions of "Basic facts about Bohr sets"

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+ | ==Ways of thinking about Bohr sets== | ||

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+ | ===Approximate subgroups=== | ||

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+ | ===Lattice convex bodies=== | ||

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+ | ===Multidimensional arithmetic progressions=== | ||

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+ | ==Converting <math>\mathbb{F}_3^n</math> concepts into <math>\mathbb{Z}_N</math> concepts== | ||

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+ | ===Subgroups go to regular Bohr sets=== | ||

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+ | ===Linear maps go to Freiman homomorphisms=== | ||

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+ | ===Linearly independent sets go to dissociated sets=== | ||

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+ | ===Codimension goes to dimension=== | ||

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+ | ===Averaging projections go to convolutions with Bohr measures=== | ||

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+ | ==Localization to a Bohr set== | ||

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+ | ===Local Fourier analysis=== | ||

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+ | ===A local Bogolyubov lemma=== | ||

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+ | ===A local Chang theorem=== |

## Revision as of 09:27, 6 February 2011

## Contents

## Definition

### Version for cyclic groups

Let [math]r_1,\dots,r_k[/math] be elements of [math]\mathbb{Z}_N[/math] and let δ>0. The *Bohr set* [math]B(r_1,\dots,r_k;\delta)[/math] is the set of all [math]x\in\mathbb{Z}_N[/math] such that [math]r_ix[/math] lies in the interval [math][-\delta N,\delta N][/math] for every i=1,2,...,k.

### Version for more general finite Abelian groups

Let G be a finite Abelian group, let [math]\chi_1,\dots,\chi_k[/math] be characters on G and let δ>0. The *Bohr set* [math]B(\chi_1,\dots,\chi_k;\delta)[/math] is the set of all [math]g\in G[/math] such that [math]|1-\chi_i(g)|\leq\delta[/math] for every i=1,2,...,k.

Note that this definition does not quite coincide with the definition given above in the case [math]G=\mathbb{Z}_N[/math]. In practice, the difference is not very important, and sometimes when working with [math]\mathbb{Z}_N[/math] it is in any case more convenient to replace the condition given by the inequality [math]|1-\exp(2\pi i r_jx/N)|\leq\delta[/math] for each j.

### Version for sets of integers

Needs to be written ...